In this context, the phrase "hour 11 of task X" means that you are in the 11th hour of doing the task. You've been doing it for somewhere between 10 and 11 hours (since the first hour goes from hour 0 to hour 1 - thanks Mathieu). The exact amount of time is not actually important to the author's point in the quote above, only that it's been a very long work day (especially at an unpleasant task).
If we instead say you performed an "11-hour task X" then we're saying that the task will be complete after 11 hours. Compare this to the "hour 11" version, which does not give any information about how long the task will have taken in total to complete (or whether it will complete at all!). The "11-hour" version also doesn't specify how much time we have already spent (only the total required) so we don't know if the task is already complete, in progress, or has not even started.
The key distinction here is that it's not a "time period" as the OP asked, but a "moment". The author chose to use this wording because he's talking about examining all of the moments in a person's life, and comparing lives which contain more pleasant (yoga) ones vs. lives that contain more unpleasant (fryer) ones. He says that if a randomly chosen moment from your life is more likely to be pleasant than unpleasant, then you are more likely to appreciate whatever the rest of the article is talking about.
It's hard to tell what exact meaning the writer was groping for. It seems pretty superfluous. I'd get rid of it, but then that's just me being my usual self. – Mick – 2016-11-28T17:24:16.960
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Very closely related! what does ''sun-dappled yoga pose''and ''hour 11 manning the deep-fat fryer'' mean?
– Mari-Lou A – 2016-11-28T17:29:44.3504"hour (number) 11 manning the deep-fat fryer” might have been easier to understand. – user3169 – 2016-11-28T17:31:16.817
@Mari-LouA it's the same user asking about a different aspect from the quote. Is this correct, or should this all have been one question? – Andrew – 2016-11-28T20:08:42.887
@Andrew the same user, but he wanted to know the meaning, I guess my explanation didn't quite convince him. He needed confirmation. However, this question is not a duplicate, it's much more specific, IOW the OP's question is different – Mari-Lou A – 2016-11-28T20:11:50.557